One of the main components in a photonic device is a waveguide or an optical fiber which serves as a low-loss medium for light transmission. An important characteristic of waveguides such as optical fiber is the dispersion that light experiences as it travels inside the waveguide. Dispersion is the phenomenon that causes different frequencies of light to travel at different velocities. The phenomenon of dispersion is commonly observed through the spreading of light by a prism. When white light, which contains a broad spectrum of frequencies, enters a prism the different wavelengths are bent at different angles since each frequency sees a different index of refraction, a phenomenon first quantified by Newton in the 17th century. Inside a fiber, this variation in the index of refraction with frequency is what causes the frequency dependence of the velocity.
A more modern example of the phenomenon of dispersion is the affect it has on the performance of photonic devices used in communication systems. In these systems, dispersion, or more specifically second order dispersion, leads to a broadening of the pulses used to represent 1 or 0 in a digital communication system. Pulse broadening causes adjacent bits to overlap and leads to intersymbol interference. Intersymbol interference occurs when a pulse is broadened beyond its allocated bit slot to such an extent that it begins to overlap with adjacent bits and it is no longer possible to determine whether or not a specific bit contains a 1 or a 0.
As a result of intersymbol interference the allocated bit slots must be widened and this effectively lowers the number of bits that can be transmitted in a given period of time and reduces the system bandwidth. As a result modern communication systems have evolved methods to mitigate the effects of dispersion.
Current methods of countering the effects of dispersion in an optical fiber use dispersion compensating devices such as chirped fiber Bragg gratings and dispersion compensating fiber (DCF). In order to effectively use these techniques it is important to know the exact magnitude of the dispersion that is being compensated for. As a result, knowledge of the dispersion in both the transmission system and the dispersion compensation system is important to the design of the overall communication system.
Knowledge of dispersion in a waveguide is also significant for the study of fiber based nonlinear wave interaction phenomena. An optical soliton is a pulse that maintains a constant shape (width) as it propagates along a fiber (first order soliton) or has a shape that is periodic with propagation (higher order soliton). This is due to the fact that the effects of dispersion and self phase modulation (SPM) are in balance. SPM is the effect whereby the phase of a given pulse is modified by its own intensity profile. Knowledge of the dispersion in an optical fiber allows for the determination of the required intensity for the formation of an optical soliton. This effect has also been used in the area of soliton effect pulse compression where the combination of the chirping effect of SPM and subsequent distributed compression effect of negative dispersion is used to compress an optical pulse. Knowledge of dispersion is also important for the study of nonlinear effects such as second harmonic generation, three-wave mixing and four-wave mixing since it determines the interaction lengths between the various wavelengths. Dispersion is particularly important in techniques that aim to extend this interaction length such as in Quasi Phase Matching (QPM) devices.
Theory on Chromatic Dispersion of a Waveguide
Dispersion is the phenomenon whereby the index of refraction of a material varies with the frequency or wavelength of the radiation being transmitted through it. The term ‘Chromatic Dispersion’ is often used to emphasize this wavelength dependence. The total dispersion in a waveguide or an optical fiber is a function of both the material composition (material dispersion) and the geometry of the waveguide (waveguide dispersion). This section outlines the contributions of both material and waveguide dispersion, identifies their physical source and develops the mathematical terminology for their description.
Dispersion in a Waveguide
When light is confined in a waveguide or an optical fiber the index is a property of both the material and the geometry of the waveguide. The waveguide geometry changes the refractive index via optical confinement by the waveguide structure. The refractive index is therefore a function of both the material and waveguide contributions. For this reason in a fiber or a waveguide the index is known as an effective index.
The dispersion parameter, D, which represents second order dispersion since it describes how the second derivative of the effective index varies with respect to wavelength:
                              D          ⁡                      (                          λ              o                        )                          =                                            -                                                λ                  o                                c                                      ⁢                                                            ⅆ                  2                                ⁢                                  n                  eff                                                            ⅆ                                  λ                  2                                                              ⁢                      |                          λ              o                                                          Eq        .                                  ⁢        1            
The dispersion parameter is important since it is related to pulse broadening which greatly limits the bit rate of a communication system.
The dispersion parameter of a waveguide such as an optical fiber is given by the total dispersion due to both the material and waveguide contributions. The total dispersion is the combination of the material dispersion and the waveguide dispersion and thus the dispersion parameter of a waveguide is given by:
                    D        =                                            -                                                2                  ⁢                  π                  ⁢                                                                          ⁢                  c                                                  λ                  2                                                      ⁢                          ⅆ                              ⅆ                ω                                      ⁢                          (                              1                                  V                  G                                            )                                =                                    D              M                        +                          D              W                                                          Eq        .                                  ⁢        2            
The next two sections discuss the contributions that both material and waveguide dispersion make individually to the total dispersion.
Material Dispersion
Material dispersion originates from the frequency or wavelength dependent response of the atoms/molecules of a material to electromagnetic waves. All media are dispersive and the only non-dispersive medium is a vacuum. The source of material dispersion can be examined from an understanding of the atomic nature of matter and the frequency dependent aspect of that nature. Material dispersion occurs because atoms absorb and re-radiate electromagnetic radiation more efficiently as the frequency approaches a certain characteristic frequency for that particular atom called the resonance frequency.
When an applied electric field impinges on an atom it distorts the charge cloud surrounding that atom and induces a polarization that is inversely proportional to the relative difference between the frequency of the field and the resonance frequency of the atom. Thus the closer the frequency of the electromagnetic radiation is to the atoms resonance frequency the larger the induced polarization and the larger the displacement between the negative charge cloud and the positive nucleus.
The material dispersion is then determined by taking the derivative of the group index of the material with respect to wavelength or equivalently the second derivative of the absolute index with respect to wavelength:
                              D          M                =                                            1              c                        ⁢                                          ⅆ                                  N                  G                                                            ⅆ                λ                                              =                                    -                              λ                c                                      ⁢                          (                                                                    ⅆ                    2                                    ⁢                  n                                                  ⅆ                                      λ                    2                                                              )                                                          Eq        .                                  ⁢        3            Waveguide Dispersion
Waveguide dispersion occurs because waveguide geometry variably affects the velocity of different frequencies of light. More technically, waveguide dispersion is caused by the variation in the index of refraction due to the confinement of light in an optical mode. Waveguide dispersion is a function of the material parameters of the waveguide such as the normalized core-cladding index difference, Δ=(ncore−ncladding)/ncore and geometrical parameters such as the core size, a. The index in a waveguide is known as an effective index, neff, because of the portion of the index change caused by propagation in a confined medium.
Confinement is best described by a quantity known as the V parameter, which is a function of both the material and geometry of the waveguide. The V parameter is given by Eq. 4:
                              V          ⁡                      (            λ            )                          =                                                            2                ⁢                π                            λ                        ⁢                                          a                ⁡                                  (                                                            n                      core                      2                                        -                                          n                      cladding                      2                                                        )                                                            1                /                2                                              ≈                                                    2                ⁢                π                            λ                        ⁢                          an              core                        ⁢                                          2                ⁢                Δ                                                                        Eq        .                                  ⁢        4            
Propagation in a waveguide is described by a quantity known as the normalized propagation constant, b, which is also a function of the material and geometry of the waveguide. This quantity is given in Eq. 5:
                    b        =                                            n              eff                        -                          n              cladding                                                          n              core                        -                          n              cladding                                                          Eq        .                                  ⁢        5            
The contribution of the waveguide to the dispersion parameter depends on the confinement and propagation of the light in a waveguide and hence it is a function of both the V parameter and the normalized propagation constant, b. The waveguide dispersion can be calculated via knowledge of V and b via Eq. 6:
                              D          W                =                  -                                                    2                ⁢                π                                            λ                2                                      ⁡                          [                                                                                          N                                              G                        ⁡                                                  (                          cladding                          )                                                                    2                                                                                      n                        cladding                                            ⁢                      ω                                                        ⁢                                                            V                      ⁢                                                                        ⅆ                          2                                                ⁢                                                  (                          Vb                          )                                                                                                            ⅆ                                              V                        2                                                                                            +                                                                            ⅆ                                              N                                                  G                          ⁡                                                      (                            cladding                            )                                                                          2                                                                                    ⅆ                      ω                                                        ⁢                                                            ⅆ                                              (                        Vb                        )                                                                                    ⅆ                      V                                                                                  ]                                                          Eq        .                                  ⁢        6            
In most cases the main effect of the waveguide dispersion in standard single mode fibers is a reduction in dispersion compared to dispersion in bulk. In comparison to material dispersion the contribution of waveguide dispersion is quite small and in most standard single mode fibers it only shifts the zero dispersion wavelength from 1276 nm to 1310 nm.
In summary, the dispersion in a waveguide or an optical fiber is caused not only by the material but also by the effect of confinement and propagation in the waveguide. Thus accurate knowledge of the dispersion in a waveguide cannot be made by simple knowledge of the material dispersion but must include the effect of the waveguide. As a result either the dimensions of the waveguide must be known to a high degree of accuracy so that the waveguide dispersion can be calculated (which is not easy since fabrication processes are hardly perfect) or the dispersion must be measured empirically. Accurate measurement of the (total) dispersion parameter, D, is important to the design of photonic systems.
Polarization Mode Dispersion
In addition to the above, optical waveguides may suffer from polarization mode dispersion (PMD). PMD may exist in fibers with asymmetrical cores. In optical fibers, the light that travels along one of the two polarization axis travels at a right angle to light traveling along the other axis. In asymmetrical optical fibers, the light travels along the two axes at different speeds. This causes pulses to spread, which can cause them to become undetectable at the detector.Conventional Measurement Techniques
There are 3 categories of dispersion measurement techniques: Time of flight (TOF), Modulation phase shift (MPS) and Interferometric. TOF and MPS are the most widely used commercial dispersion measurement techniques. Interferometric techniques are not widely used commercially but have been used in laboratories for dispersion measurements. Interferometric techniques come in two forms; temporal and spectral. The existing techniques differ in measurement precision and fiber length requirements.
Time of Flight Technique
In the TOF technique the second order dispersion parameter, D, hereafter referred to simply as the dispersion parameter, can be determined either by measuring the relative temporal delay between pulses at different wavelengths or by measuring the pulse broadening itself. The relative temporal delay between pulses at different wavelengths is measured to determine the group velocity which can then be used to determine the dispersion parameter using Eq. 7:
                              D          ⁡                      (                          λ              o                        )                          =                              Δ            ⁢                                                  ⁢            t                                L            ⁢                                                  ⁢            Δ            ⁢                                                  ⁢                          λ              ⁡                              (                                  λ                  o                                )                                                                        Eq        .                                  ⁢        7            
The above equation can also be used to determine the dispersion parameter from the pulse broadening itself if Δt is the measured pulse broadening and Δλ is the bandwidth of the wavelengths in the pulse. The measurement precision achievable by the TOF technique is on the order of 1 ps/nm.
One of the main problems with the TOF technique is that it generally requires several kilometers of fiber to accumulate an appreciable difference in time for different wavelengths. Another issue with the TOF technique when the pulse broadening is measured directly is that the pulse width is affected by changes in the pulse shape which leads to errors in the measurement of the dispersion parameter. As a result, in order to measure the dispersion parameter with a precision near 1 ps/nm-km several kilometers of fiber are required.
Modulation Phase Shift Technique
The MPS technique is another dispersion characterization technique that requires long lengths of fiber. In the MPS technique, a continuous-wave optical signal is amplitude modulated by an RF signal, and the dispersion parameter is determined by measuring the RF phase delay experienced by the optical carriers at the different wavelengths.
The RF phase delay information is extracted by this technique, and by taking the second derivative of the phase information, the dispersion parameter can be determined. Measurement precision achievable by the MPS technique is on the order of 0.07 ps/nm. Due to its higher precision, MPS has become the industry standard for measuring dispersion in optical fibers. However, MPS has several disadvantages. The first is that it is expensive to implement since the components required such as an RF analyzer and a tunable laser, are costly. The second is that its precision is limited by both the stability and jitter of the RF signal.
MPS has several limitations on the minimum device length that it is capable of characterizing. In the MPS method the width of the modulated signal limits the minimum characterizable device length. This method also typically requires fiber lengths in excess of tens of meters to obtain a precision to better than 1 ps/nm-km. Therefore it is not desirable for the characterization of specialty fibers or gain fibers, of which long fiber lengths are expensive to acquire or not available. Also, when fiber uniformity changes significantly along its length, the dispersion of a long span of fiber cannot be used to accurately represent that of a short section of fiber. In such cases, dispersion measurement performed directly on short fiber samples is desirable. As a result a technique for measuring the dispersion of short lengths of fiber is desired.
Dispersion Measurements on Short Length
Interferometric techniques are capable of characterizing the dispersion on fiber lengths below 1 m. There are two categories of interferometric techniques for making dispersion measurements on fiber of short length: temporal and spectral.
Temporal Interferometry (Fourier Transform Spectroscopy)
Dual Arm temporal interferometry employs a broadband source and a variable optical path to produce a temporal interferogram between a fixed path through the test fiber and variable air path. It involves moving one arm of the interferometer at a constant speed and plotting the interference pattern as a function of delay length (time). The spectral amplitude and phase are then determined from the Fourier transform of the temporal interferogram.
A temporal interferogram gives the phase variation as a function of time. The spectral phase variation can be extracted from the temporal interferogram if a Fourier Transform is applied to it. The spectral phase contains the dispersion information which can be indirectly obtained by taking the second derivative of the spectral phase. A precision of 0.0015 ps/nm measured on a 0.814-m-long photonic crystal fiber was recently reported using temporal interferometry. The main disadvantage of temporal interferometry is that it is susceptible to noise resulting from both translation inaccuracy and vibration of the optics in the variable path. A tracking laser is typically required to calibrate the delay path length. Another problem with this technique is that a second derivative of the phase information must be taken to obtain the dispersion parameter which means that it is less accurate than a method that can obtain the dispersion parameter directly.
Spectral Interferometry
Spectral interferometry, like temporal interferometry, is capable of characterizing the dispersion in short length fiber (<1 m). In spectral interferometry, instead of stepping the length of one of the arms, a scan of the wavelength domain performed to produce a spectral interferogram. Spectral interferometry is generally more stable than temporal interferometry since the arms of the interferometer are kept stationary. Thus it is simpler than temporal interferometry since no tracking laser is necessary.
There are two types of spectral interferometry, one is general and does not require balancing, and another, the special case, is ‘balanced’. In the balanced case it is possible to directly measure the dispersion parameter from the interferogram. This makes it more accurate than temporal interferometry and it is for this reason that spectral interferometry is discussed as a dispersion measurement technique.
In general spectral interferometry the dispersion parameter is obtained from the interference spectrum produced by two time delayed light pulses/beams in an unbalanced dual arm interferometer. Two pulses/beams from the two arms of the interferometer are set up to interfere in a spectrometer and a spectral interferogram is produced. The interference pattern produced for a given time or phase delay is given by:
                                                                        I                ⁡                                  (                  ω                  )                                            =                            ⁢                                                                                                                                    E                        o                                            ⁡                                              (                        ω                        )                                                              +                                                                  E                        (                                                 ⁢                        ω                        )                                            ⁢                                              exp                        ⁡                                                  (                                                      ⅈω                            ⁢                                                                                                                  ⁢                            τ                                                    )                                                                                                                                      2                                                                                        =                            ⁢                                                                                                                                      E                        o                                            ⁡                                              (                        ω                        )                                                                                                  2                                +                                                                                                E                      ⁡                                              (                        ω                        )                                                                                                  2                                +                                                                            E                      o                      *                                        ⁡                                          (                      ω                      )                                                        ⁢                  E                  ⁢                                                                          ⁢                                      ω                    ⁡                                          (                      ω                      )                                                        ⁢                                      exp                    ⁡                                          (                      ⅈωτ                      )                                                                      +                                                                                                      ⁢                                                                    E                    o                                    ⁡                                      (                    ω                    )                                                  ⁢                                                      E                    *                                    ⁡                                      (                    ω                    )                                                  ⁢                                  exp                  ⁡                                      (                                          -                      ⅈωτ                                        )                                                                                                                          =                            ⁢                                                                                                                                      E                        o                                            ⁡                                              (                        ω                        )                                                                                                  2                                +                                                                                                E                      ⁡                                              (                        ω                        )                                                                                                  2                                +                                                      f                    ⁡                                          (                      ω                      )                                                        ⁢                                      exp                    ⁡                                          (                      ⅈωτ                      )                                                                      +                                                                                                      ⁢                                                                    f                    *                                    ⁡                                      (                    ω                    )                                                  ⁢                                  exp                  ⁡                                      (                                          -                      ⅈωτ                                        )                                                                                                          Eq        .                                  ⁢        8            
The last two terms in Eq. 8 result in spectral interference pattern via a cos(Δφ(ω)+ωτ) term.
There are several ways to extract the phase information from the cosine term but the most prevalent way to do so is to take the Inverse Fourier transform of the spectral interference pattern. Note that f(ω)=F.T.f(t)=|E*o(ω)E(ω)|exp(iΔφ(ω)) contains all the phase information on the spectral phase difference Δφ(ω). Therefore, if f(ω) can be extracted from the interference pattern then the phase difference information can be known. If an Inverse Fourier Transform of the spectral interference is performed on the interference pattern the following is obtained:F.T.−1(I(ω))=E*o(−t){circumflex over (x)}Eo(t)+E*(−t){circumflex over (x)}E(t)+f(t−τ)+f(−t−τ)  Eq. 9
If all terms except the f(t−τ) term get filtered out via a band pass filter then the phase information can be extracted from a Fourier Transform on f(t−τ).
The phase information can then be extracted by applying a Fourier Transform to the filtered component f(t−τ) thereby transferring it back to the spectral domain. The complex amplitude therefore becomes f(ω)=|Eo(ω)∥E(ω)|exp(iΔφ(ω)+ωr). The phase of this complex amplitude minus the linear part (ωτ) that is due to the delay, yields the spectral phase difference between the two pulses as a function of ω and is independent of the delay between the two pulses. In this way the phase difference between the two pulses can be obtained.
If one of the pulses travels through a non-dispersive medium such as air and the other pulse travels through a dispersive medium such as an optical fiber then the phase difference spectrum will be directly related to the dispersion in the fiber. Thus the dispersion parameter plot can be determined by taking the second derivative of the phase difference spectrum with respect to wavelength.
The main issue with this form of spectral interferometry, however, is that the dispersion parameter is not determined directly but rather via a second order derivative of the phase information with respect to wavelength. Therefore, like temporal interferometry, this general unbalanced method of spectral interferometry is not as accurate as the balanced method capable of measuring the dispersion parameter directly of the present invention.
In balanced spectral interferometry the arm lengths of an interferometer are kept constant and they are balanced for a given wavelength called the central wavelength such that the group delay in both arms is the same. This allows for the removal of the effect of the large linear dispersion term in the interferogram. Balanced interferometry measures the dispersion parameter D at the wavelength at which the group delay is the same in both arms. This wavelength is henceforth referred to as the central wavelength. The dispersion parameter in this case can be directly determined from the phase information in the spectral interferogram without differentiation of the phase. For this reason it is more accurate than both unbalanced general spectral interferometry and temporal interferometry. As a result balanced spectral interferometry is often used to obtain accurate dispersion measurements in short length waveguides and fibers.
Both forms of spectral interferometry are considered to be less susceptible to noise since the arms of the interferometer are kept still and there are no moving parts. It is for this reason that spectral interferometry in general is considered to be more accurate than temporal interferometry. Spectral interferometry is therefore considered to be the technique of choice for measuring the dispersion of photonic components and spectral depth resolved optical imaging. One well known and important class of spectral interferometry is optical coherence tomography (OCT).
Balanced dual arm spectral interferometers are typically found in a Michelson or a Mach-Zehnder configuration in which the path lengths are equalized at the given wavelength in which the dispersion is to be measured. The most often used configuration, however, is the Michelson and the discussion that follows considers the Michelson interferometer. In a balanced Michelson interferometer the dispersion is measured from the interference between two waves: one that passes through the test fiber and another that passes through an air path. Balancing the air path length with the fiber eliminates the effect of the group index of the fiber in the interference pattern. This allows for the measurement of the second derivative of the effective index with respect to wavelength directly from the interference pattern.
The main disadvantage of this Michelson configuration is that two types of path balancing must occur simultaneously. The first type of path balancing is coupler arm balancing wherein the coupler arms need to be balanced exactly or an extra set of interference fringes will be created from the reflections at the two end facets of the coupler arms as shown in FIG. 1.
The second type of balancing is test fiber-air path balancing to ensure that the optical path length in the air path exactly equals that of the fiber for a given central wavelength. This ensures that the central wavelength in the interference pattern is within the viewable bandwidth of the OSA.
The main problem in implementing a Michelson interferometer is that the arms of the coupler cannot be balanced exactly and as a result the effect of the extra set of reflections produced at the coupler facets cannot be removed.
Comparison of Dispersion Measurement Techniques
There have been several techniques developed for the measurement of chromatic dispersion in fiber. Especially important are those developed for the measurement of short lengths of fiber. One reason that short length characterization techniques are important stems from recent developments in the design and fabrication of specialty fiber.
Specialty fiber such as Twin Hole Fiber (THF) (FIG. 14) and Photonic Crystal Fiber (PCF) have made short length fiber characterization desirable due to their high cost. Because of this it is not economical to use TOF and MPS techniques to characterize these types of fiber. Another impetus for short length characterization comes from the fact that in many specialty fibers the geometry is often non-uniform along its length. As a result of this non-uniformity the dispersion in these fibers varies with position. Thus measurement of the dispersion in a long length of this fiber will be different than that measured in a section of the same fiber.
Based on the above discussion, the technique of choice for dispersion measurement is balanced spectral interferometry since it will provide the most accurate measurements. As a result the new technique will employ balanced spectral interferometry.
The two important parameters in comparing dispersion measurement techniques are the minimum device length that each is capable of characterizing and the precision to which the characterization is achieved. It is generally desirable to characterize as short a fiber as possible with as high a precision as possible. It is also desirable to perform the measurement in the simplest way possible.
Therefore, what is needed is a new method for the measurement of dispersion that does not require the cancellation of any extra fringes. What is also needed is a method to measure the dispersion parameter in short lengths of optical fiber. The initial need for a short length characterization scheme came from the need to measure the dispersion of a specialty fiber such as THF, PCF, or gain fibre. This requirement is based on the expense of fibre, nonlinear wave interaction phenomena in fibre, and non-uniform dispersion along the length of a fibre.